Problem 6 - Applied Calculus

Question

The following table gives the percentage, \(P\), of households with cable television between 1977 and \(2003 .^{15}\) $$\begin{array}{c|r|r|r|r|r|r|l}\hline \text { Year } & 1977 & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 \\\\\hline P & 16.6 & 17.9 & 19.4 & 22.6 & 28.3 & 35.0 & 40.5 \\\\\hline \text { Year } & 1984 & 1985 & 1986 & 1987 & 1988 & 1989 & 1990 \\\\\hline P & 43.7 & 46.2 & 48.1 & 50.5 & 53.8 & 57.1 & 59.0 \\ \hline \text { Year } & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 \\\\\hline P & 60.6 & 61.5 & 62.5 & 63.4 & 65.7 & 66.7 & 67.3 \\\\\hline \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & \\\\\hline P & 67.4 & 68.0 & 67.8 & 69.2 & 68.9 & 68.0 & \\\\\hline\end{array}$$ (a) Explain why a logistic model is reasonable for this data. (b) Estimate the point of diminishing returns. What limiting value \(L\) does this point predict? Does this limiting value appear to be accurate, given the percentages for 2002 and \(2003 ?\) (c) If \(t\) is in years since \(1977,\) the best fitting logistic function for this data turns out to be $$P=\frac{68.8}{1+3.486 e^{-0.237 t}}$$ What limiting value does this function predict? (d) Explain in terms of percentages of households what the limiting value is telling you. Do you think your answer to part (c) is an accurate prediction?

Step-by-Step Solution

Answer

A logistic model is reasonable due to slowing growth rate. The estimated limiting value is 68.8%, which aligns with the data trend from 2000-2003.

Step 1: Understanding Logistic Models

A logistic model is reasonable for this data because logistic growth describes a situation where growth starts exponentially but decelerates as it approaches a limiting value. The data shows rapid growth in cable television adoption that gradually slows as penetration increases, suggesting it is approaching a saturation point or limiting value.

Step 2: Calculating the Point of Diminishing Returns

The point of diminishing returns in a logistic model occurs when the rate of growth begins to slow significantly. Observing the table, the growth rate decreases notably after the year 1990 (59%). This indicates approaching the point of diminishing returns. The limiting value appears to be around 68-70% as it fluctuates near this level from 2000 to 2003.

Step 3: Using the Logistic Function to Find Limiting Value

Given the logistic function: \[ P=\frac{68.8}{1+3.486 e^{-0.237 t}} \] As \( t \to \infty \), the exponential term \( e^{-0.237 t} \to 0 \), so \[ P = \frac{68.8}{1+0} = 68.8 \]. Therefore, the function predicts a limiting value of 68.8%.

Step 4: Interpreting the Limiting Value

The limiting value of 68.8% represents the predicted maximum penetration of cable television in households. It is the percentage at which almost all households that can have cable do have it. Observing the data from 2000 to 2003, this prediction is reasonable as the percentage hovers around 68%, suggesting saturation is near this value.

Key Concepts

Point of Diminishing Returns

In the context of a logistic growth model, the point of diminishing returns represents the stage at which growth starts to decelerate. Initially, a rapid increase is observed, but over time, the growth rate begins to slow. The data on cable television adoption shows a classic display of this behavior. Initially, from 1977 to 1990, there was a rapid increase in adoption rates. But post-1990, when the percentage reached about 59%, the growth rate began to taper off significantly. This pattern is indicative of the point of diminishing returns, where the initially exponential growth moves towards a more gradual incline.

Reaching this point means that most of the potential for rapid growth has been exhausted. For cable television, it suggests that the remaining households are adopting cable at a slower rate. This point is crucial for identifying when a significant change in growth dynamics occurs, and it is a hallmark of logistic growth models in capturing saturation behavior in a market.

The observed data, especially in the years following 1990, reflects typical logistic growth, where predicting further growth becomes challenging due to the slowing increase and proximity to the limiting value.

Limiting Value

The limiting value in a logistic growth model is the theoretical maximum to which the growth can approach but never truly exceed. It represents the saturation point, where adding more of the variable of interest (in this case, cable television) becomes challenging because most of the potential market has been covered.

From the step-by-step solution, using the logistic function, the limiting value was calculated as 68.8%. This limiting value means that it is predicted that 68.8% is the highest percentage of households that will eventually have cable television.

This theoretical limit is supported by the data from 2000 to 2003, where the percentage fluctuates close to this value. Factors like market saturation, consumer preferences, and technological alternatives can impose such limits. In practical scenarios, reaching close to the limiting value indicates that almost all of the market demand has been met. When planning for the future, businesses use this value to forecast when and where they may need to innovate or diversify.

Logistic Function Analysis

The logistic function is a significant tool in modeling growth that is initially exponential but approaches a limiting value. For cable television household adoption, the logistic function is given by \[ P=\frac{68.8}{1+3.486 e^{-0.237 t}} \]This equation helps determine future predictions about the growth trend. As time \(t\) increases, the exponential part of the function \( e^{-0.237 t} \) approaches zero, simplifying the equation to \( P = 68.8 \).

Here, 68.8 represents the limiting value, suggesting that P (percentage of households with cable) will close in on 68.8% as time progresses. This type of function analysis confirms the tendency toward a saturation point based on historical data, aligning with the predicted market capacity.

Understanding and correctly applying logistic function analysis enable prediction not only in statistical modeling but also in other fields such as biology, economics, and technology adoption. It helps in planning, strategy formulation, and understanding when new interventions might be needed due to diminishing returns and market saturation.